Unified representation of the family of L -functions

Hacer Ozden; Yilmaz Simsek

doi:10.1186/1029-242X-2013-64

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Published: 21 February 2013

Unified representation of the family of L-functions

Hacer Ozden¹ &
Yilmaz Simsek²

Journal of Inequalities and Applications volume 2013, Article number: 64 (2013) Cite this article

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Abstract

The aim of this paper is to unify the family of L-functions. By using the generating functions of the Bernoulli, Euler and Genocchi polynomials, we construct unification of the L-functions. We also derive new identities related to these functions. We also investigate fundamental properties of these functions.

AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.

1 Introduction

The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.

Throughout this presentation, we use the following standard notions $N = {1, 2, …}$ , $N 0 = {0, 1, 2, …} = N ∪ {0}$ , $Z + = {1, 2, 3, …}$ , $Z − = {− 1, − 2, …}$ . Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real number and ℂ denotes the set of complex numbers. We assume that $ln (z)$ denotes the principal branch of the multi-valued function $ln (z)$ with the imaginary part $ℑ (ln (z))$ constrained by $− π < ℑ (ln (z)) ≤ π$ .

Recently, the first author [1] introduced and investigated the following generating functions which give a unification of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials:

g a, b (x; t, k, β) : = 2 1 − k t k e t x β b e t − a b = ∑ n = 0 ∞ Y n, β (x; k, a, b) t n n!,

(1)

where ( $| t | < 2 π$ when $β = a$ ; $| t | < | b log (β a) |$ when $β ≠ a$ ; $k ∈ N 0$ ; $β ∈ C$ ( $| β | < 1$ ); $a, b ∈ C ∖ {0}$ ).

For the special values of a, b, k, b and β, the polynomials $Y n, β (x; k, a, b)$ provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.

Remark 1.1 If we set $k = a = b = 1$ in (1), we get a special case of the generalized Bernoulli polynomials $Y n, β (x, k, 1, 1)$ , that is, the so-called Apostol-Bernoulli polynomials $B n (x, β)$ generated by

t β e t − 1 e x t = ∑ n = 0 ∞ B n (x, β) t n n!

(cf. [1–28]).

Remark 1.2 By substituting $k + 1 = − a = b = 1$ in (1), we are led to Apostol-Euler polynomials $E n (x, β)$ which are defined by means of the following generating function:

2 β e t + 1 e x t = ∑ n = 0 ∞ E n (x, β)

(cf. [1–28]).

Remark 1.3 Setting $k = − a = b = 1$ into (1), we get the Apostol-Genocchi polynomials $G n (x, β)$ which are defined by means of the following generating function:

2 t β e t + 1 e x t = ∑ n = 0 ∞ G n (x, β) t n n!

(cf. [1–28]).

In terms of a Dirichlet character χ of conductor $f ∈ N$ , Ozden et al. [16] extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.

Definition 1.4 (Ozden et al. [[16], p.2783])

Let χ be a Dirichlet character of conductor $f ∈ N$ . Then the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi numbers $Y n, χ, β (k, a, b)$ and the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi polynomials $Y n, χ, β (x; k, a, b)$ are given by the following generating functions:

F χ, β (t; k, a, b) = 2 1 − k t k ∑ j = 1 f χ (j) (β a) b j e j t β b f e f t − a b f = ∑ n = 0 ∞ Y n, χ, β (k, a, b) t n n!,

(2)

where ( $| t | < 2 π$ when $β = a$ ; $| t | < | b log (β a) |$ when $β ≠ a$ ; $k ∈ N 0$ ; $β ∈ C$ ( $| β | < 1$ ); $a, b ∈ C ∖ {0}$ ) and

H χ, β (x, t; k, a, b) = F χ, β (t, k; a, b) e t x = ∑ n = 0 ∞ Y n, χ, β (x; k, a, b) t n n!

(3)

( $| t | < 2 π$ when $β = a$ ; $| t | < | b log (β a) |$ when $β ≠ a$ ; $k ∈ N 0$ ; $β ∈ C$ ( $| β | < 1$ ); $a, b ∈ C ∖ {0}$ ).

Remark 1.5 Substituting $k = a = b = β = 1$ into (2), we are led immediately to the generating function of the generalized Bernoulli numbers which are defined by means of the following generating function:

∑ j = 1 f χ (j) t e j t e f t − 1 = ∑ n = 0 ∞ B n, χ t n n!

(4)

(cf. [1–26]).

2 Unification of the L-functions

Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials $Y n, χ, β (x; k, a, b)$ in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate $Y n, χ, β (x; k, a, b)$ for negative integer values of n.

Throughout this section, we assume that $β ∈ C$ with $| β | < 1$ and $s ∈ C$ .

By substituting (1) into (2), we obtain the following functional equation:

F χ, β (t; k, a, b) = 1 f k ∑ j = 1 f χ (j) (β a) b j g a f, b (j f, t f; k, β f) .

(5)

By using this functional equation, we arrive at the following theorem.

Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have

Y n, χ, β (k, a, b) = f n − k ∑ j = 1 f χ (j) (β a) b j Y n, β f (j f; k, a f, b) .

(6)

By using (5), we modify (3) as follows:

H χ, β (x, t; k, a, b) = 1 f k ∑ j = 1 f χ (j) (β a) b j g a f, b (j + x f, t f; k, β f) .

(7)

By using (7), we derive the following result.

Corollary 2.2 Let χ be a Dirichlet character of conductor $f ∈ N$ . Then we have

Y n, χ, β (x; k, a, b) = f n − k ∑ j = 1 f χ (j) (β a) b j Y n, β f (j + x f; k, a f, b) .

(8)

By applying the Mellin transformation to the generating function (1), Ozden et al. [[16], p.2784 Equation (4.1)] gave an integral representation of the unified zeta function $ζ β (s, x; k, a, b)$ :

ζ β (s, x; k, a, b) = 1 Γ (s) ∫ 0 ∞ t s − k − 1 g a, b (x; − t; k, β) d t (min {ℜ (s), ℜ (x)} > 0),

(9)

where the additional constraint $ℜ (x) > 0$ is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [[16], p.2784 Equation (4.1)] defined the unified zeta function $ζ β (s, x; k, a, b)$ as follows:

ζ β (s, x; k, a, b) = (− 12) k − 1 ∑ m = 0 ∞ β b m a b (m + 1) (m + x) s (β ∈ C (| β | < 1); s ∈ C (ℜ (s) > 1)) .

(10)

By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions $L χ, β (s, x; k, a, b)$ :

(11)

in terms of the generating function $H χ, β (x, t; k; a, b)$ defined in (7). By substituting (9) into ( 11), we obtain

L χ, β (s, x; k, a, b) = 1 f k + s ∑ j = 1 f χ (j) (β a) b j ζ β f (s, j + x f; k, a f, b)

(12)

where ( $β ∈ C$ ( $| β | < 1$ ); $s ∈ C$ ( $ℜ (s) > 1$ )).

Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions $L χ, β (s, x; k, a, b)$ as follows.

Definition 2.3 Let χ be a Dirichlet character of conductor $f ∈ N$ . For $s, β ∈ C$ ( $| β | < 1$ ), we define a two-variable unified L-function $L χ, β (s, x; k, a, b)$ by

L χ, β (s, x; k, a, b) = f − k (− 12) k − 1 ∑ m = 0 ∞ β b m χ (m) a b (m + f) (m + x) s (β ∈ C (| β | < 1); ℜ (s) > 1) .

(13)

Remark 2.4 If we substitute $x = 1$ into (13), we get the unified L-function

L χ, β (s; k, a, b) : = L χ, β (s, 1; k, a, b)

by

L χ, β (s; k, a, b) = f − k (− 12) k − 1 ∑ m = 1 ∞ β b m χ (m) a b (m + f) m s,

where ( $ℜ (s) > 1$ , $β ∈ C$ ( $| β | < 1$ )).

Remark 2.5 Upon substituting $k = a = b = 1$ and $β = ξ u$ into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:

l 1 (u ξ, s, χ) = L χ, ξ u (s, x; 1, 1, 1),

where, for a positive integer r, ξ is the r th root of 1.

l 1 (u ξ, s; χ) = ∑ m = 0 ∞ (ξ u) m χ (m) (m + x) s

(cf. [18]).

Remark 2.6 Substituting $x = 1$ into (13), we get a unification of the L-functions

L χ, β (s, 1; k, a, b) = L χ, β (s; k, a, b) .

Substituting $χ ≡ 1$ into (13), we get a unification $ζ β (s, x; k, a, b)$ of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function

ζ (s, x) = ζ 1 (s, x; 1, 1, 1) = ∑ n = 0 ∞ 1 (n + x) s

(cf. [27, 28]) and the Riemann zeta function

ζ (s) = ζ 1 (s, 1; 1, 1, 1) = ∑ n = 1 ∞ 1 n s

are obvious special cases of the unified zeta function $ζ β (s, x; k, a, b)$ (cf. [16, 27, 28]). The relationship between the unified zeta function and the Hurwitz-Lerch zeta function $Φ (z, s, a)$ was given by Ozden et al. [16]:

ζ β (s, x; k, a, b) : = (− 12) k − 1 a − b Φ (β b a b, s, x),

(14)

where the Hurwitz-Lerch zeta function is defined by

Φ (z, s, x) = ∑ n = 0 ∞ z n (n + x) s,

which converges for ( $x ∈ C ╲ Z 0 −$ , $s ∈ C$ when $| z | < 1$ ; $ℜ (s) > 1$ when $| z | = 1$ ), where as usual

Z 0 − = Z − ∪ {0}

(cf. [27, 28]).

A relationship between the functions $L χ, β (s, x; k, a, b)$ and $ζ β (s, x; k, a, b)$ is provided by the next theorem.

Theorem 2.7 Let $s ∈ C$ . Let χ be a Dirichlet character of conductor $f ∈ N$ . Then we have

L χ, β (s, x; k, a, b) = f − s − k ∑ j = 1 f (β a) j b χ (j) ζ β f (s, j + x f; k, a f, b) .

(15)

Proof Substituting $m = n f + j$ , $j = 1, 2, …, f$ , $n = 0, …, ∞$ into (13), we obtain

L χ, β (s, x; k, a, b) = (− 12) k − 1 f − s − k ∑ j = 1 f (β a) j b χ (j) ∑ n = 0 ∞ β b n f a b n f (n + j + x f) s .

After some algebraic manipulations, we arrive at the desired result. □

Remark 2.8 Substituting $a = b = k = 1$ into (13), we have

L χ, β (s, x; 1, 1, 1) = ∑ m = 0 ∞ β m χ (m) (m + x) s (ℜ (s) > 1, β ∈ C (| β | < 1))

which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:

∑ j = 1 f χ (j) t β j e t (j + x) β f e t f − 1 = ∑ n = 0 ∞ B n, χ (x, β) t n n! .

Let f be an odd integer. If we set $a = − 1$ and $k = 0$ into (13), then we have

L χ, β (s, x; 1, − 1, 1) = 2 ∑ m = 1 ∞ (− 1) m χ (m) β m (m + x) s (ℜ (s) > 1, β ∈ C (| β | < 1)),

which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:

∑ j = 1 f 2 χ (j) β j e t (j + x) β f e t f + 1 = ∑ n = 0 ∞ E n, χ (x, β) t n n!

(cf. [1–29]).

By using (15) and (14), we arrive at the following result.

Corollary 2.9 Let $s ∈ C$ . Let χ be a Dirichlet character of conductor $f ∈ N$ . Then we have

L χ, β (s, x; k, a, b) = (− 12) k − 1 a − f b f − s − k ∑ j = 1 f (β a) j b χ (j) Φ (β f b a f b, s, j + x f) .

Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have

L χ, β (1 − n, x; k, a, b) = (− 1) k f (n − 1)! (n + k − 1)! Y n + k − 1, χ, β (x; k, a, b) .

(16)

Proof By substituting $s = 1 − n$ into (15), we get

L χ, β (1 − n, x; k, a, b) = f n − 1 − k ∑ j = 1 f (β a) j b χ (j) ζ β f (1 − n, j + x f; k, a f, b) .

By using Theorem 7 in [16], we get

By substituting (8) into the above, we arrive at the desired result. □

Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions $L χ, β (s, x; k, a, b)$ defined by (13). We thus have (cf. [13])

L (s, x; χ) = ∑ m = 0 ∞ χ (m) (m + x) s

and

L (s; χ) = ∑ m = 1 ∞ χ (m) m s,

where $ℜ (s) > 1$ . By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have

L (1 − n; χ) = − B n, χ n,

where $n ∈ Z +$ and $B n, χ$ , the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that $L (s; χ)$ is non-zero at $s = 1$ . Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at $s = 1$ (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).

3 Applications

In this section, by using (16) and the following formula, which was proved by Ozden et al. [[16], Theorem 5, Equation (3.10)]

Y n, χ, β (x; k, a, b) = ∑ j = 0 n (n j) x n − j Y j, χ, β (k, a, b),

(17)

we construct a meromorphic function involving a unified family of L-functions. Therefore, using (16) and (17),

L χ, β (1 − n, x; k, a, b) = x n + k − 1 f ∏ l = 0 k − 1 (n + l) ∑ j = 0 n + k − 1 (n + k − 1 j) 1 x j Y j + k − 1, χ, β (k, a, b) .

From the above equation, we arrive at the following theorem.

Theorem 3.1 Let $x ≠ 0$ . Let χ be a Dirichlet character of conductor f. Then we have

L χ, β (s, x; k, a, b) = x k − s f ∏ l = 0 k − 1 (s − 1 − l) ∑ j = 0 ∞ (k − s j) 1 x j Y j + k − 1, χ, β (k, a, b) .

The function $L χ, β (s, x; k, a, b)$ is an analytic function at $s = 0$ . We now compute the value of this function at this point as follows:

L χ, β (0, x; k, a, b) = x k (− 1) k f ∏ l = 0 k − 1 (1 + l) ∑ j = 0 k (k j) 1 x j Y j + k − 1, χ, β (k, a, b) .

The function $L χ, β (s, x; k, a, b)$ is a meromorphic function. This function has simple poles which are

s = 1, 2, 3, …, k .

The residues of this function at the simple poles at $s = 1$ and $s = k$ are given, respectively, as follows:

Res s = 1 {L χ, β (s, x; k, a, b)} = x k − 1 f (− 1) k ∏ l = 0 k − 1 (2 + l) ∑ j = 0 k − 1 (k − 1 j) 1 x j Y j + k − 1, χ, β (k, a, b)

and

Res s = k {L χ, β (s, x; k, a, b)} = Y k − 1, χ, β (k, a, b) f ∏ l = 0 k − 2 (k − 1 − l) .

Remark 3.2 Simsek (cf. [20, 21]) defined a twisted two-variable L-function $L ξ, q (h) (s, x; χ)$ as follows:

L ξ, q (h) (s, x; χ) = ∑ m = 0 ∞ χ (m) ϕ ξ (m) q h m (x + m) s − log q h s − 1 ∑ m = 0 ∞ χ (m) ϕ ξ (m) q h m (x + m) s − 1,

where $q ∈ C$ ( $| q | < 1$ ); $ξ r = 1$ ( $r ∈ Z$ ); $ξ ≠ 1$ . Observe that if $ξ = 1$ , then $L ξ, q (h) (s, x; χ)$ is reduced to the work of Kim [9].

Relationship between the function $L χ, β (s, x; k, a, b)$ and $L ξ, q (h) (s, x; χ)$ is given as the following result.

Corollary 3.3 Let χ be a Dirichlet character of conductor f. Then we have

L 1, β b a b (b) (s, x; χ) = (− 2) k a b f f k (L χ, β (s, x; k, a, b) − log q h s − 1 L χ, β (s − 1, x; k, a, b)) .

We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function $Φ (z, s, a)$ , Hurwitz zeta function $ζ (s, x)$ and L-function (cf. [1–30]); see also the references cited in each of these earlier works.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

Both authors are partially supported by Research Project Offices Akdeniz Universities and the Commission of Scientific Research Projects of Uludag University Project number UAP(F) 2011/38 and 2012/16. We would like to thank referees for their valuable comments.

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Department of Mathematics, Faculty of Art and Science, University of Uludag, Bursa, Turkey
Hacer Ozden
Department of Mathematics, Faculty of Science, Akdeniz University, Campus, Antalya, 07058, Turkey
Yilmaz Simsek

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Ozden, H., Simsek, Y. Unified representation of the family of L-functions. J Inequal Appl 2013, 64 (2013). https://doi.org/10.1186/1029-242X-2013-64

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Received: 05 December 2012
Accepted: 04 February 2013
Published: 21 February 2013
DOI: https://doi.org/10.1186/1029-242X-2013-64

Keywords

Bernoulli numbers
Bernoulli polynomials
Euler numbers
Euler polynomials
Genocchi numbers
Genocchi polynomials
Dirichlet L-functions
Hurwitz zeta function
Riemann zeta function